Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo

We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in $L^p$ and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the $\operatorname {GARCH}(1,1)$ Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.